1. Field of the Invention
The present invention relates to a multiplier circuit for multiplying two input signals and more particularly, to a bipolar analog multiplier capable of perfect four-quadrant multiplication operation by using a multitail cell as a multiplier core circuit, which is preferably formed on a bipolar semiconductor integrated circuit (IC), and which is operable at a low supply voltage.
2. Description of the Related Art
A typical example of the conventional bipolar analog multipliers is the "Gilbert multiplier cell" shown in FIG. 1, which was disclosed in IEEE Journal of Solid-State Circuits, Vol. SC-3, No. 4, pp. 353-365, December, 1968, entitled "A Precise Four quadrant Analog Multiplier with Subnanosecond Response", and written by B. Gilbert.
In FIG. 1, npn bipolar transistors Q901 and Q902 form a first emitter-coupled differential pair, npn bipolar transistors Q903 and Q904 form a second emitter-coupled differential pair, and npn bipolar transistors Q907 and Q908 form a third emitter-coupled differential pair.
Collectors of the transistors Q901, Q902, Q903 and Q904 are cross-coupled. A collector of the transistor Q907 is connected to the coupled emitters of the transistors Q901 and Q902. A collector of the transistor Q908 is connected to the coupled emitters of the transistors Q903 and Q904. The coupled emitters of the transistors Q907 and Q908 are connected to a constant current sink sinking a constant current I.sub.0. Bases of the transistors Q901 and Q904 are coupled together. Bases of the transistors Q902 and Q903 are also coupled together.
A first input signal voltage V.sub.x is applied across the coupled bases of the transistors Q901 and Q904 and those of the transistors Q902 and Q903. A second input signal voltage V.sub.y is applied across the bases of the transistors Q907 and Q908.
The third differential pair of the transistors Q907 and Q908 and the corresponding constant current sink constitute a differential voltage-current (V-I) converter for the voltage V.sub.y
A collector current of the transistor Q907 is expressed as (I.sub.0 /2)+(I.sub.y /2)!, and a collector current of the transistor Q908 is expressed as (I.sub.0 /2)-(I.sub.y /2)!, where I.sub.y is a collector current generated by the input voltage V.sub.y.
An output current I.sup.+ is derived from the coupled collectors of the transistors Q901 and Q903, and another output current I.sup.- is derived from the coupled collectors of the transistors Q902 and Q904. A differential output current .DELTA.I of the Gilbert multiplier cell containing the multiplication result of the first and second input signal voltages V.sub.x and V.sub.y is obtained by the difference of the two output currents I.sup.+ and I.sup.- ; i.e., .DELTA.I=I.sup.+ -I.sup.-.
The differential output current .DELTA.I is expressed as ##EQU1## where V.sub.T is the thermal voltage defined as V.sub.T =kT/q, where k is the Boltzmann's constant, T is absolute temperature in degrees Kelvin, and q is the charge of an electron.
When V.sub.x .ltoreq.V.sub.T and V.sub.y .ltoreq.V.sub.T, the differential output current .DELTA.I is approximated as ##EQU2##
The well-known Gilbert multiplier of FIG. 1 is unable to realize the perfect four-quadrant multiplication operation, which is due to the hyperbolic tangent (tanh) characteristic of the cross-coupled, emitter-coupled differential pairs of the transistors Q901, Q902, Q903, and Q904 and the nonlinear operation of the V-I converter formed by the transistors Q907 and Q908.
FIG. 2 shows a conventional analog multiplier realizing the perfect four-quadrant multiplication operation. This multiplier has the same cross-coupled, emitter-coupled differential pairs formed by the transistors Q901, 0902, 0903, and Q904 as those in the Gilbert multiplier cell of FIG. 1
Instead of the V-I converter formed by the transistors Q907 and Q908 in FIG. 1, a perfect-linear V-I converter 973 is provided. An arc hyperbolic tangent (tanh.sup.-1) converter 971 and a perfect-linear V-I converter 972 are additionally provided.
The tanh.sup.-1 converter 971 is formed by diode-connected npn bipolar transistors Q905 and Q906, and the coupled bases and collectors of the transistors Q905 and Q906 are connected to a power supply (supply voltage: V.sub.cc). The converter 971 serves as a p-n junction element.
The first input signal voltage V.sub.x is applied across the input terminals of the V-I converter 972, and then, is converted to a pair of differential output currents I.sub.x.sup.+ and I.sub.x.sup.-. The differential output currents I.sub.x.sup.+ and I.sub.x.sup.- are then tanh.sup.-1 -converted by the tanh.sup.-1 converter 971, thereby generating a differential output voltage .DELTA.V.sub.x at the emitters of the transistors Q905 and Q906.
The differential output voltage .DELTA.V.sub.x is proportional to tanh.sup.-1 of the first input signal voltage V.sub.x. The voltage .DELTA.V.sub.x is applied across the coupled bases of the transistors Q901 and Q904 and those of the transistors Q902 and Q903.
Since the applied voltage .DELTA.V.sub.x is proportional to tanh.sup.-1 of the first input signal voltage V.sub.x, the tanh characteristic of the cross-coupled, emitter-coupled pair formedby the transistors Q901, Q902, Q903, and Q904 is compensated, resulting in a perfect-linear operation with respect to the first input signal voltage V.sub.x.
On the other hand, the second input signal voltage V.sub.y is applied across the perfect-linear V-I converter 973, and then, is linearly converted to a pair of differential output currents I.sub.y.sup.+ and I.sub.y.sup.- ; The cross-coupled, emitter-coupled pairs formed by the transistors Q901, Q902, QD03, and Q904 are driven by the pair of differential output currents I.sub.y.sup.+ and I.sub.y.sup.-. Accordingly, the operation of the cross-coupled, emitter-coupled pairs become linear with respect to the second input signal voltage V.sub.y.
As a result, the perfect four-quadrant multiplication operation can be realized with respect to both of the first and second input signals V.sub.x and V.sub.y. This means that the four-quadrant multiplier capable of perfect-linear operation can be realized.
The perfect-linear V-I converters 972 and 973 are termed "linear transconductance amplifiers" or "linear gain cells".
Next, the circuit operation of the conventional multiplier of FIG. 2 is explained below.
Supposing that the base-width modulation (i.e., the Early voltage) is ignored, a collector current Ic of abipolar transistor is typically expressed as the following equation (3) based on the exponential-law characteristic. ##EQU3## where V.sub.BE is the base-to-emitter voltage of the transistor, and I.sub.s is the saturation current thereof.
In the equation (3), the term of exp(V.sub.BE /V.sub.T) has a value of approximately e.sup.10 during the normal operation of a bipolar transistor when the base-to-emitter voltage V.sub.BE is approximately 600 mV. Therefore, the term of (-1) can be ignored.
Thus, the equation (3) is approximated to the following equation (4). ##EQU4##
In the following analysis, for the sake of simplification, it is supposed that the common-base current gain factor of the transistor is approximately equal to unity and therefore, the base current can be ignored.
In the V-I converter 972, the following equations (5) and (6) are established. ##EQU5## where V.sub.BE905 and V.sub.BE906 are the base-to-emitter voltages of the transistors Q905 and Q906, respectively, and 2G.sub.x is the conductance of the V-I converter 972 (i.e., I.sub.x.sup.+ -I.sub.x.sup.- =2G.sub.x V.sub.x).
Accordingly, the differential output voltage .DELTA.V.sub.x of the converter 971 is given by the following equation (7). ##EQU6##
On the other hand, the differential output current .DELTA.I of the multiplier in FIG. 2 is expressed as the following equation (8). ##EQU7##
It is seen from the equation (8) that the differential output current .DELTA.I is proportional to the tanh of the differential input voltage .DELTA.V.sub.x.
The equation (8) is obtained by using the equation (7) and the following identity (9). ##EQU8##
The difference of the pair of differential output currents I.sub.y.sup.+ and I.sub.y.sup.-, i.e., (I.sub.y.sup.+ -I.sub.y.sup.-) in the equation (8) is expressed as ##EQU9##
The expression (10) is obtained by using the following identity (11). ##EQU10##
Thus, the differential output current .DELTA.I in the equation (8) is rewritten to the following expression (12). ##EQU11##
The expression (12) shows that the conventional multiplier of FIG. 2 is capable of the perfect four-quadrant multiplication operation with respect to both of the first and second input signals V.sub.x and V.sub.y. In other words, it can be said that the conventional multiplier of FIG. 2 is a "translinear multiplier".
An analog multiplier is an essential, basic function block in analog signal applications. Recently, the need for an analog multiplier capable of perfect four-quadrant multiplication operation, which is linear for the two input signal voltages, has been increased.